求解双曲型守恒律的半离散三阶中心迎风格式

给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD Runge—Kutta方法。本文格式保持了中心差分格式简单的优点,即不需用Riemann解算器,避免了进行特征分解过程。用该格式对一维和二维守恒律进行了大量的数值试验,结果表明本文格式是高精度、高分辨率的。     

Adaptive multi-resolution central-upwind schemes for systems of conservation laws

In this article, we develop an adaptive scheme for solving systems of hyperbolic conservation laws. In this scheme nonlinear shock and linear contact waves will be treated differently. The proposed scheme uses the Kurganov central-upwind scheme. Fourth-order non-oscillatory reconstruction is employed near shock only while the unlimited fifth-order reconstruction is used for smooth regions and linear contact waves. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multi-resolution technique. The advantages of the scheme are high resolution and computationally efficient since limiters are used only for shocks. Numerical experiments with one- and two-dimensional problems are presented which show the robustness of the proposed scheme.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

In this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd.     

求解双曲守恒律方程的WENO型熵相容格式

通过在单元交界面处进行高阶WENO重构,得到了一种求解双曲型守恒律方程的WENO型熵相容格式。用该格式对一维Burgers方程和Euler方程进行数值模拟,结果表明,该格式具有高精度、基本无振荡性等特点。

     

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws

A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourth-order central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments. The project supported by the National Natural Science Foundation of China (60134010) The English text was polished by Yunming Chen.     

An improved third-order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L_∞ and L₂ norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes.     

非线性双曲型守恒律的两步二阶TVD差分格式

通过Mac Cormack格式和Warming-Beam的结合,构造了一种非常简单的两步二阶TVD差分格式,该差分格式更适合于使用分量形式差分计算而无须对欧拉方程组进行特征解耦。通过对流体力学方程组的大量数值试验,并与二阶ENO格式进行了比较,充分显示了该格式高精度、高分辨并且极其简单的优良特性。     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

A high‐resolution hybrid scheme for hyperbolic conservation laws

A new hybrid scheme is proposed, which combines the improved third‐order weighted essentially non‐oscillatory (WENO) scheme presented in this paper with a fourth‐order central scheme by a novel switch. Two major steps have been gone through for the construction of a high‐performance and stable hybrid scheme. Firstly, to enhance the WENO part of the hybrid scheme, a new reference smoothness indicator has been devised, which, combined with the nonlinear weighting procedure of WENO‐Z, can drive the third‐order WENO toward the optimal linear scheme faster. Secondly, to improve the hybridization with the central scheme, a hyperbolic tangent hybridization switch and its efficient polynomial counterpart are devised, with which we are able to fix the threshold value introduced by the hybridization. The new hybrid scheme is thus formulated, and a set of benchmark problems have been tested to verify the performance enhancement. Numerical results demonstrate that the new hybrid scheme achieves excellent performance in resolving complex flow features, even compared with the fifth‐order classical WENO scheme and WENO‐Z scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Methods for extending high-resolution schemes to non-linear systems of hyperbolic conservation laws

In extending high-resolution methods from the scalar case to systems of equations there are a number of options available. These options include working with either conservative or primitive variables, characteristic decomposition, two-step methods, or component-wise extension. In this paper, several of these options are presented and compared in terms of economy and solution accuracy. The characteristic extension with either conservative or primitive variables produces excellent results with all the problems solved. Using primitive variables, the two-step formulation produces high-quality results in a more economical manner. This method can also be extended to multiple dimensions without resorting to dimensional splitting. Proper selection of limiters is also important in non-characteristic extension to systems.     

Improved shock-capturing of Jameson's scheme for the Euler equations

It is known that Jameson's scheme is a pseudo-second-order-accurate scheme for solving discrete conservation laws. The scheme contains a non-linear artificial dissipative flux which is designed to capture shocks. In this paper, it is shown that the, shock-capturing of Jameson's scheme for the Euler equations can be improved by replacing the Lax-Friedrichs' type of dissipative flux with Roe's dissipative flux. This replacement is at a moderate expense of the calculation time.     

Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric

Approximate symmetries have been defined in the context of differential equations and systems of differential equations. They give approximately, conserved quantities for Lagrangian systems. In this paper, the exact and the approximate symmetries of the system of geodesic equations for the Schwarzschild metric, and in particular for the radial equation of motion, are studied. It is noted that there is an ambiguity in the formulation of approximate symmetries that needs to be clarified by consideration of the Lagrangian for the system of equations. The significance of approximate symmetries in this context is discussed.     

An implicit characteristic-flux-averaging method for the euler equations for real gases

A formulation of an implicit characteristic-flux-averaging method for the unsteady Euler equations with real gas effects is presented. Incorporation of a real gas into a general equation of state is achieved by considering the pressure as a function of density and specific internal energy. The Ricmann solver as well as the flux-split algorithm are modified by introducing the pressure derivatives with respect to density and internal energy. Expressions for calculating the values of the flow variables for a real gas at the cell faces are derived. The Jacobian matrices and the eigenvectors are defined for a general equation of state. The solution of the system of equations is obtained by using a mesh-sequencing method for acceleration of the convergence. Finally, a test case for a simple form of equation of state displays the differences from the corresponding solution for an ideal gas.     

On the relation between the entropy balance and the numerical solutions of systems of conservation laws

This work deals with the relation between the numerical solutions of hyperbolic systems of conservation laws and the associated entropy evolution. An analysis of the continuum problem by means of variational calculus clearly emphasizes the consequences of the adopted reconstruction procedure on the induced entropy balance. A methodology is proposed that allows for a posterior local and global spurious entropy production estimates on the basis of an additional equation representing a discrete approximation to the entropy inequality. The problem of defining a consistent approximation of the numerical entropy flux is also addressed in detail. Properly designed numerical experiments support the analysis and contribute to providing a more comprehensive evaluation of the numerical entropy dynamics. © 1997 John Wiley & Sons, Ltd.     

基于四阶半离散中心迎风格式的虚拟流方法的应用

给出了求解多维无粘可压Euler方程组的四阶半离散中心迎风格式,该格式根据非线性波在网格单元边界上传播的局部速度来更准确地估计局部Riemann的宽度,避免了计算网格的交错,降低了格式的数值粘性。同时,考虑到Level Set函数能隐式地追踪到界面的位置,而虚拟流的构造能隐式地捕捉到界面的边界条件,因此再将新的四阶半离散中心迎风格式与Level Set方法以及虚拟流方法相结合,成功地处理了非反应激波和多介质流中爆轰间断的追踪问题。     

Derivative artificial compression method for conservation laws with multi-dimensional extension

A new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.     

Undercompressive shocks for nonstrictly hyperbolic conservation laws

We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.